Wiley signals and systems e book TLFe BO 310

In Chapters 1to 10 we gcjt to know wmc powertul tools for dealirig with contiiiuous signals arid systems We carried out the convei sion of red-world coritiiiiioirs signals t o sanipled signals in Chapter I, which is necessary foi digital processing The sampled signals were tieated as conitiniioiis-tiinevariables so thal the rools we had learnt , like the Foiirier transforin, could A l l br iised In a coiiipiater, we can only work with a sequence of niuriberi, that are defined over a discrete range by a numrrical value (the index) Examples of such an index might be the S ~ ~ I T ~nuniber P~C fox a digital aiidio signal, or the pixel address within a digital image In addition there are discrete sigrials t h t haw not arisen from sampling a continuous signal Example:, of these are represented in Figurc~1.3 and 1.4 W e need iiew tools to descrihe such secpien , arid w e will concentrate or1 them in Chsptcrs 12 - 14 This chapter dcals with dzscrrte signulr and the discrete form of the Foiirier transtorm, the J, trun.$fm-m This traiisforni is also rcferrcd to as dhwrete-tme Fourqrr traiLcsfOrm (DTFT) The two following (liaptrrs deal with dzscretr systems and we will leain the discrete counterpart to the Lsplace transform , the z-lrunsform In Swtions 12.1 arid 12 wtl will coiisider diicrete signals together with sonic exaniples The discrete-time Fowler transform will also be introduced, which we will iisc to exaniirie discrete signals in the frcqueiicy-domain We will sec that i t has sirriilai properties to the Foiarier transform for coiitinuous sign& At the end of this chapter we will investigate the relationship between coiitiiiiioiis signals and their disrrete cquivaleiit as a series of samples 42.11 A discrete-time signal is represented by a sequence of iiuriihcrs that is callrd a time series There is no sniooth transition between the nurnl-jers Figure 12.1 shows the conventions we will use to reprrsent such signals; in order to distinguish them from continuous-time signal$, we put the independent vaf iable in square brackt In inany technical appiications, a cliscrete-time signal arises from the sarnplzrry of a continuoiis-tiiii~ signal ( i ) , wherc a sample is taken kern P ( t ) at regular